Theses and Dissertations from UMD
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Item Adaptive Sampling for Geometric Approximation(2020) Abdelrazek, Ahmed Abdelkader; Mount, David M; Computer Science; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Geometric approximation of multi-dimensional data sets is an essential algorithmic component for applications in machine learning, computer graphics, and scientific computing. This dissertation promotes an algorithmic sampling methodology for a number of fundamental approximation problems in computational geometry. For each problem, the proposed sampling technique is carefully adapted to the geometry of the input data and the functions to be approximated. In particular, we study proximity queries in spaces of constant dimension and mesh generation in 3D. We start with polytope membership queries, where query points are tested for inclusion in a convex polytope. Trading-off accuracy for efficiency, we tolerate one-sided errors for points within an epsilon-expansion of the polytope. We propose a sampling strategy for the placement of covering ellipsoids sensitive to the local shape of the polytope. The key insight is to realize the samples as Delone sets in the intrinsic Hilbert metric. Using this intrinsic formulation, we considerably simplify state-of-the-art techniques yielding an intuitive and optimal data structure. Next, we study nearest-neighbor queries which retrieve the most similar data point to a given query point. To accommodate more general measures of similarity, we consider non-Euclidean distances including convex distance functions and Bregman divergences. Again, we tolerate multiplicative errors retrieving any point no farther than (1+epsilon) times the distance to the nearest neighbor. We propose a sampling strategy sensitive to the local distribution of points and the gradient of the distance functions. Combined with a careful regularization of the distance minimizers, we obtain a generalized data structure that essentially matches state-of-the-art results specific to the Euclidean distance. Finally, we investigate the generation of Voronoi meshes, where a given domain is decomposed into Voronoi cells as desired for a number of important solvers in computational fluid dynamics. The challenge is to arrange the cells near the boundary to yield an accurate surface approximation without sacrificing quality. We propose a sampling algorithm for the placement of seeds to induce a boundary-conforming Voronoi mesh of the correct topology, with a careful treatment of sharp and non-manifold features. The proposed algorithm achieves significant quality improvements over state-of-the-art polyhedral meshing based on clipped Voronoi cells.Item Accurate Data Approximation in Constrained Environments(2005-06-15) Deligiannakis, Antonios; Roussopoulos, Nick; Computer Science; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Several data reduction techniques have been proposed recently as methods for providing fast and fairly accurate answers to complex queries over large quantities of data. Their use has been widespread, due to the multiple benefits that they may offer in several constrained environments and applications. Compressed data representations require less space to store, less bandwidth to communicate and can provide, due to their size, very fast response times to queries. Sensor networks represent a typical constrained environment, due to the limited processing, storage and battery capabilities of the sensor nodes. Large-scale sensor networks require tight data handling and data dissemination techniques. Transmitting a full-resolution data feed from each sensor back to the base-station is often prohibitive due to (i) limited bandwidth that may not be sufficient to sustain a continuous feed from all sensors and (ii) increased power consumption due to the wireless multi-hop communication. In order to minimize the volume of the transmitted data, we can apply two well data reduction techniques: aggregation and approximation. In this dissertation we propose novel data reduction techniques for the transmission of measurements collected in sensor network environments. We first study the problem of summarizing multi-valued data feeds generated at a single sensor node, a step necessary for the transmission of large amounts of historical information collected at the node. The transmission of these measurements may either be periodic (i.e., when a certain amount of measurements has been collected), or in response to a query from the base station. We then also consider the approximate evaluation of aggregate continuous queries. A continuous query is a query that runs continuously until explicitly terminated by the user. These queries can be used to obtain a live-estimate of some (aggregated) quantity, such as the total number of moving objects detected by the sensors.